54.1.347 problem 354

Internal problem ID [11661]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 354
Date solved : Tuesday, September 30, 2025 at 10:02:21 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (x \sin \left (y\right )-1\right ) y^{\prime }+\cos \left (y\right )&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 108

ode:=(x*sin(y(x))-1)*diff(y(x),x)+cos(y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \arctan \left (\frac {-\sqrt {c_1^{2}-x^{2}+1}\, c_1 +x}{c_1^{2}+1}, \frac {c_1 x +\sqrt {c_1^{2}-x^{2}+1}}{c_1^{2}+1}\right ) \\ y &= \arctan \left (\frac {\sqrt {c_1^{2}-x^{2}+1}\, c_1 +x}{c_1^{2}+1}, \frac {c_1 x -\sqrt {c_1^{2}-x^{2}+1}}{c_1^{2}+1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.686 (sec). Leaf size: 163

ode=Cos[y[x]] + (-1 + x*Sin[y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\arccos \left (\frac {c_1 x-\sqrt {-x^2+1+c_1{}^2}}{1+c_1{}^2}\right )\\ y(x)&\to \arccos \left (\frac {c_1 x-\sqrt {-x^2+1+c_1{}^2}}{1+c_1{}^2}\right )\\ y(x)&\to -\arccos \left (\frac {\sqrt {-x^2+1+c_1{}^2}+c_1 x}{1+c_1{}^2}\right )\\ y(x)&\to \arccos \left (\frac {\sqrt {-x^2+1+c_1{}^2}+c_1 x}{1+c_1{}^2}\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 1.858 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x*sin(y(x)) - 1)*Derivative(y(x), x) + cos(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = 2 \operatorname {atan}{\left (\frac {\sqrt {C_{1}^{2} - x^{2} + 1} - 1}{C_{1} - x} \right )}, \ y{\left (x \right )} = - 2 \operatorname {atan}{\left (\frac {\sqrt {C_{1}^{2} - x^{2} + 1} + 1}{C_{1} - x} \right )}\right ] \]